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To derive the Ginzburg criterion for the upper-critical dimension the fluctuations are averaged over a volume set by the correlation length. Why is this done? i.e. why do we average over a volume in the first place and why when we do not the entire volume?

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  • $\begingroup$ The field theory itself is obtained by averaging local degrees of freedom over scales smaller than the correlation length. See Kardar's book Statistical Physics of Fields for an explanation of all this, and the RG flow, etc. $\endgroup$ – Ryan Thorngren Apr 17 '18 at 9:49
  • $\begingroup$ A possible answer to this question along the lines of @RyanThorngren's comment is given in Goldenfeld, 2006; pg148 and 169 $\endgroup$ – Quantum spaghettification Apr 20 '18 at 14:19
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Here is a simple way to think about it: Consider the Feynman diagrammatic expansion of the statistical field theory. The propagator for the order parameter field is $1/(p^2+m^2)$, where $m=1/\xi$ is the inverse correlation length. We can now study the scaling behavior of diagrams as follows $$ \int \frac{1}{p^2+m^2} \sim \int_m^\Lambda \frac{1}{p^2} $$ where $\Lambda$ is a UV cutoff. In coordinate space $\xi=1/m$ becomes an IR (long distance) cutoff. This corresponds to averaging fluctuations over a volume of size $V\sim \xi^d$.

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Mean field theory ignores fluctuations of the field and is valid if these fluctuations are smaller then mean field contribution. You thus need to sum over all fluctuations or in real space you need to consider fluctuations between any two parts of the system. Since the correlation length sets the length scale in which you know fluctuations have decayed to 0. You in fact only need to integrate over a volume set by the correlation length. This is what is expressed by the ~ in Thomas's response.

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